Grant Todd
01/16/2024 · High School
Use integration by parts to evaluate the integral: \( \int \tan ^{-1} x d x \)
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Step-by-step Solution
Calculate the integral \( \int \tan ^{-1} x d x \).
Evaluate the integral by following steps:
- step0: Evaluate using partial integration formula:
\(\int \arctan\left(x\right) dx\)
- step1: Prepare for integration by parts:
\(\begin{align}&u=\arctan\left(x\right)\\&dv=dx\end{align}\)
- step2: Calculate the derivative:
\(\begin{align}&du=\frac{1}{1+x^{2}}dx\\&dv=dx\end{align}\)
- step3: Evaluate the integral:
\(\begin{align}&du=\frac{1}{1+x^{2}}dx\\&v=x\end{align}\)
- step4: Substitute the values into formula:
\(\arctan\left(x\right)\times x-\int \frac{1}{1+x^{2}}\times x dx\)
- step5: Calculate:
\(x\arctan\left(x\right)-\int \frac{x}{1+x^{2}} dx\)
- step6: Evaluate the integral:
\(x\arctan\left(x\right)-\frac{1}{2}\ln{\left(x^{2}+1\right)}\)
- step7: Add the constant of integral C:
\(x\arctan\left(x\right)-\frac{1}{2}\ln{\left(x^{2}+1\right)} + C, C \in \mathbb{R}\)
The integral of \( \int \tan ^{-1} x d x \) using integration by parts is \( x\arctan(x)-\frac{1}{2}\ln{(x^{2}+1)} + C \), where \( C \) is an arbitrary constant.
Quick Answer
\( x\arctan(x)-\frac{1}{2}\ln{(x^{2}+1)} + C \)
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